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Combinatorial Group Testing in Presence of Deletions

Venkata Gandikota, Nikita Polyanskii, Haodong Yang

TL;DR

The paper addresses non-adaptive combinatorial group testing under adversarial deletions, a noise model that causes test outcome shifts and asynchrony. It introduces deletion-separable and deletion-disjunct matrices to characterize when defectives can be uniquely recovered after up to $\Delta$ deletions, and proves both necessary and sufficient conditions. The authors provide randomized constructions achieving $m = O(k^2\log n + \Delta k)$ tests with efficient decoding, along with a sublinear-time decoding variant inspired by SAFFRON and an explicit KS-like deterministic construction using codes with strong deletion-distance properties. These results advance robust CGT in deletion-prone channels and data storage settings, offering practical decoding algorithms and near-optimal test counts within current theoretical limits. The work opens several avenues for explicit, improved constructions and deeper code-theoretic developments in insertion-deletion contexts.

Abstract

The study in group testing aims to develop strategies to identify a small set of defective items among a large population using a few pooled tests. The established techniques have been highly beneficial in a broad spectrum of applications ranging from channel communication to identifying COVID-19-infected individuals efficiently. Despite significant research on group testing and its variants since the 1940s, testing strategies robust to deletion noise have yet to be studied. Many practical systems exhibit deletion errors, for instance, in wireless communication and data storage systems. Such deletions of test outcomes lead to asynchrony between the tests, which the current group testing strategies cannot handle. In this work, we initiate the study of non-adaptive group testing strategies resilient to deletion noise. We characterize the necessary and sufficient conditions to successfully identify the defective items even after the adversarial deletion of certain test outputs. We also provide constructions of testing matrices along with an efficient recovery algorithm.

Combinatorial Group Testing in Presence of Deletions

TL;DR

The paper addresses non-adaptive combinatorial group testing under adversarial deletions, a noise model that causes test outcome shifts and asynchrony. It introduces deletion-separable and deletion-disjunct matrices to characterize when defectives can be uniquely recovered after up to deletions, and proves both necessary and sufficient conditions. The authors provide randomized constructions achieving tests with efficient decoding, along with a sublinear-time decoding variant inspired by SAFFRON and an explicit KS-like deterministic construction using codes with strong deletion-distance properties. These results advance robust CGT in deletion-prone channels and data storage settings, offering practical decoding algorithms and near-optimal test counts within current theoretical limits. The work opens several avenues for explicit, improved constructions and deeper code-theoretic developments in insertion-deletion contexts.

Abstract

The study in group testing aims to develop strategies to identify a small set of defective items among a large population using a few pooled tests. The established techniques have been highly beneficial in a broad spectrum of applications ranging from channel communication to identifying COVID-19-infected individuals efficiently. Despite significant research on group testing and its variants since the 1940s, testing strategies robust to deletion noise have yet to be studied. Many practical systems exhibit deletion errors, for instance, in wireless communication and data storage systems. Such deletions of test outcomes lead to asynchrony between the tests, which the current group testing strategies cannot handle. In this work, we initiate the study of non-adaptive group testing strategies resilient to deletion noise. We characterize the necessary and sufficient conditions to successfully identify the defective items even after the adversarial deletion of certain test outputs. We also provide constructions of testing matrices along with an efficient recovery algorithm.
Paper Structure (15 sections, 17 theorems, 19 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 15 sections, 17 theorems, 19 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation
  4. Setup
  5. Preliminaries
  6. Our Results
  7. Construction of $(k, \Delta)$ - Deletion Separable Matrix & Decoding
  8. Decoding Algorithm
  9. Deletion disjunct matrix & decoding
  10. Decoding Algorithm
  11. Case 1: ($i \in \textsc{supp}(x^*)$):
  12. Case 2: ($i \notin \textsc{supp}(x^*)$):
  13. Deletion Disjunct Matrix with Sublinear Time Decoding
  14. Deterministic construction of $(k,\Delta)$-deletion disjunct matrix
  15. Conclusion and Open Problems

Key Result

Theorem 5

There exists a $(k, \Delta)$ - deletion separable matrix $A \in \{0,1\}^{m \times n}$ with $m = O(\Delta \cdot k^2\log{n})$.

Figures (1)

  • Figure 1: Testing Matrix Construction

Theorems & Definitions (40)

  • Definition 1: Separable Matrix
  • Definition 2: Disjunct Matrix
  • Definition 3
  • Definition 4: $(k, \Delta)$-deletion separable matrix
  • Theorem 5
  • proof : Proof of Theorem \ref{['thm:del-dist']}
  • Definition 6: Asymmetric deletion distance
  • Definition 7: $(k, \Delta)$-deletion disjunct matrix
  • Lemma 8
  • proof
  • ...and 30 more